Gear and gear tooth



Sept. 11, 1934. N. TRBOJEVICH GEAR AND GEAR TOOTH Filed Aug. 14, 1931 4Sheets-Sheet l INVENTOR gem-.4

ATTORNEYS Sept. 11, 1934. N. TVRBOJEVICH 1,973,185

GEAR AND GEAR TOOTH Filed Aug. 14, 19:51 4 Sheets-Sheet 2 INVENTORATTORNEYS Sept. 11, 1934. N ATRBQJEVICH 1,973,185

GEAR AND GEAR TOOTH Filed Aug. 1.4, 1951 4 Sheets-Sheet 3 INVENTOR I BY21/ ATTORNEYS Sept. 11, 1934. u N. TRBOJEVICH 1,973,185

GEAR AND GEARwTOOTH Filed Aug. 14, 1931 4 Sheets-Sheet 4 44m xiii 233 34BY W I ATTORNEY;

Patented Sept. 11, 1934 uNrrs STATES PATENT orrr.

The invention relates to a novel system of tooth curves and isapplicable to spur, helical, bevel and worm gears and racks.

The object of my invention is so toiorin the toothcurves that therelative radii of curvature in the mating faces will, be considerablyincreased and the I-Iertz stresses thereby reduced.

Heretofore it was customary to generate a mating pinion gear fromvthesame basicrack with the result that the pinion flanks were sharplycurved and, therefore, highly stressed under load. I conceived the ideaof generating the pinion from a rack P the flanks of which are convexand the (larger) mating gear from a rack G having concave faces. By aproper selection of the rack flank curvature I amnow enabled toconstruct a pinion and gear in which the radii of curvature in thepinion are increased at the expense of the gearradii. The principle isof a particular value for high ratios of transmission, 1. e., in whichthe diameters of the mating gears greatly differ.

The objects of this invention are to produce quieter gears to increasethe permissible loading, and to equalize the surface stresses and thedepthwise deformations in various phases of engage,- ment.

In the drawings:

Figure 1 is a diagram showing the effect of th rack curvature upon thegenerated curve;

Figures 2 and 3 show the two new basic racks P and G, complementary toeach other;

Figure 4 shows the new line of action employed in this gearing;

Figure 5 shows a rack P in engagement with my improved pinion;

Figure 6 shows a rack G in engagement with my improved gear;

Figure '7 is a diagram used in connection with the equations 1am 20;

Figures 8 and9 are diagrams explaining the principle upon which theareas of vmaximum stress are avoided in my construction;

Figure 10 shows a pinion and gear of the new design in mesh; 7 I tFigure 11 illustrates the method of bobbing two mating helical gears onthis principle;

Figures 12land l3 diagrammatically show an improved worm andworrn gearof this kind; and

Figures 14 and 15show two cutters of the face mill type to generatespiral bevel pinions and gears respectively according to this principle.

In the following discussion I shall use the terincurvat1. .re in itsstrictly numerical sense meaning the reciprocal of the radiusofcurvature. In two mating curves, theirelative curvature willbe the sumof the particular curvatures, the convexfaces being considered as of apositive and the concave faces as of anegative curvature."

In the theory of gearing I think I am the first: to discover thefollowing rule: For given center distance, ratio and pressure angletherelative curvature of two mating curves is @011: stant at the pitchpoint, regardless of the curvature of the'generating rack. f a

The proof of this theorem will be seen from Figure 1. Letthe pitch line21 of the (convex) rack 23 rollover the pitch circle 22 of thegear.Upon. an infinitesimal roll from A to A the center of curvatureOz of therack will travel to 03. The center of curvature of the mating curve 24will be at the I point 04, said point being the intersection of the twoconsecutive normals 02 A and 03 A of the path curve 02 03. The are A Bstruck. from O4 is, hence, parallel to the are O2 O3 Neglecting thedifferentials of the higher order than the first I have, I

p b a sin 79 where!) is'the radius of curvature of the rack 23 and p theradius of the generated curve 24.

Equation 5' shows that the relative curvature existing between the toothcurve 24 and the rack 23 at the pitch point A is a constant. dependinguponv the pressure angle 70. rack curve, in this case, is

and to generate a curve capable of meshing with The curvature of the thecurve 24, I must use another rack having a negative curvature equal toor, inotherwords, it is possible to increase the radius of curvature ofone 1 tooth curve at the expense of the mating tooth curve by employingtwo racks, one convex and one concave, in generation.

Therefore, my system of generating possesses a flexibility which the nowknown and employed systems do not have. To illustrate with an example,let the ratio be six to one. gearing, the radius of the pinion flankwould be, then, one sixth of the gear flank radius. In my system,however, I may increase the pinion radius I at will, for instance, I canmake both radii equal} In Figures 2 and 3 the two new racks P and G areshown, the first having convex and the second concave flanks 23. Inorder to simplify the manufacture of the cutting tools such as hobs,fellows cutters and grinders, I select the curve 23 to be a true circle.This limitation is not theoretically necessary, however. I also selectthe radius of the P'rack to be slightly greater and the radius of the Grack slightly less than I) in order to ease on" the contact at the tipsand roots of the teeth.

It is to be noted thatthe racks P and G are complementary of each other,as a mechanic would say, one is male and the other female. Two gearswill mesh together ifone is out with the P and the other with the G rackor hob. But two P or two G gears will not run't-ogether.

Long addendum gears may also be generated in this system similar to theconventional system. Suppose that in the P rack the pitch line 21 islowered through a distance ,1 in the new position 25, then the pitchline of the G rack must be raised the corresponding amount. Inasmuch asother, no extra back lash will develop during such a procedure.

The line of action 26 of either the racks P or G is plotted in Figure 4.A series of normals such as 02 E F are drawn to the rack curve 23 andthe intercepts with the pitch line 21, such as F E are scaled off fromthe pitch point A by constructing a series of parallelograms, such as AF E E. The locus of the points E gives the new line of action 26. Thiscurve happens to be an already known curve, the so-called conchoid ofNicomedes whenever the rack curve 23 is a circle. Its polar equationreferred to the pitch line 21 as the axis and the point A as the originis readily found, viz.,

2 =(b+m) sin 'y (11) the same pressure angle and depth of tooth. For

this reason, the new pinions possess a better over- .lapping action andmay be constructed to have fewer teeth than was feasible, to useheretofore.

The construction of the improved pinion tooth curve 32 is shown inFigure 5. AI rack having convex circular faces 23 is rolled over thepinion 33 in such a manner that the corresponding pitch lines 21 and 28do not slip one over the other. A Relative to the fixed center 01 andthe pinion 33 the center 02 of the rack curve 23 describes an extendedinvolute having a positive modification p relative to the commoninvolute 30 developed In ordinary from the pitch circle 28. The exactvalue of the said modification is 10:17 sin 'y (13) and the equation ofthe curve 29 relative to the axes X Y is where the geometrical meaningof the parameter 4) is as in Figure 5.

Let now R denote the radius of curvature of the curve 29 such as, forinstance, the radius 02 05, then The tooth curve 32 is an envelope of afamily of circles of a radius 1) all having their respective In order toanalyze the behaviour of the curve 32 at any point thereof, I transformthe Equations 15 and 16 by means of the diagram Figure '7.

p +a =(b+m) (17 si 'y (18) sin 'y= (19) b+m= ff where 8 denotes thedistance of the point in en- 7 gagement from the pitch line of the rack.

The generation of the mating gear 34 by means of a G rack is shown inFigure 6. The path curve of the rack center 02 is now an abridgedinvolute 35 having exactly the opposite (a negative) modification p tothe former case and is derived from the common involute 36. TheEquations 13 to 20 exactly apply to the curve 35 when p is considered asa negative number. The tooth curve 37 possesses the evolute 38 in commonwith 3 the abridged involute 35 but is now situated at the opposite sideof the said evolute.

. In designing the new gears I begin with analyzing the tooth curve 3'7,Figure 6, either graphically orby means of the Equations 13 to 20. In

particular, there is a danger that the curve 37 will run into too smalla. radius near its root if the rack radius 1) or the number of teeth inthe gear is selected too small. On the other hand,

there is no diificulty whatever with the rack P for which reason mysystem is at its best in pinion and rack drives and in worm and wheeldrives. As to the rack G, to cite an example, for 6 Dia. Pitch, standarddepth of tooth, for b=3.5"

the said rack will'generate all gears from 2'7 teeth 1.

and up,20 pressure angle. 'By raising the lower limit as to the numberof teeth the radius 1) may be reduced and a more intensive and effectivemodification obtained.

radius of curvature. When two flanks are involved, both curved, then themeasure 'of stress is the momentary relative radius of curvature. Thestress will be infinite whenever the radius of curvature of either ofthe participating flanks is equal to zero. This is exactly the conditionwhich my invention avoids. In Figure 8 a conventional involute tooth 41is shown in mesh with its rack 40. The radii of curvature are developedfrom the base circle 42 and constantly increase from M to I to N, thevalue at M being substantially equal to zero. Inasmuch as the rack flank43 is a straight line, the relative radii of curvature of the twomembers 40 and 41 are exactly equal to the pinion radii at M, T and N.In Figure 9 the ordinate indicates the stress S and the abscissa thedeveloped arc M T N. As the contact travels from M towards N, the stressvaries along the curve 44. In my improved rack and pinion drive, seeFigure 5, the stresses are practically constant as represented by thecurve 45. The mean stress S0 at the pitch point A is the same in both ofthese and all other systems as was shown in the Equations 1 to 5. Thegain consists in eliminating the area of maximum stress 45a at the lefthand side of Figure 9.

I shall now briefly recapitulate this somewhat complicated development.As shown in Figure 10 the tooth curve 32 of the pinion 33 is generatedfrom the extended involute 29 drawn from the modified base radius a1+pwith a positive modification p. The process of generation of the curve32 from the curve 29 consists in sealing oil a constant distance I)along the normals of the latter, i. e., the two curves are equidistantfrom each other. Similarly the tooth curve 37 of the mating gear 34 isgenerated from the abridged involute 35 drawn from the modified baseradius a2p with a negative modification p. The process of generation ofthe curve 3'7 from the curve 35 consists in scaling oil a constantdistance b along the normals of the latter, 1. e., the two curves areequidistant from each other.

As was shown previously, this leads to the employment of two racks P andG, both having exactly the same line of action, a conchoidal spiral andthe first having convex and the second concave faces or flanks. It willbe seen that the gears thus generated will correctly mesh and with auniform velocity because the line of action is exactly the same ineither rack.

In manufacturing the new gears, hobs, fellows cutters and grinders maybe employed. In Figure 11 the principle of hobbing helical gears of thistype is illustrated. A right hand pinion 46 meshes with a left hand gear47. Two hobs are used to generate this pair, the hob 48 of the P type(having convex faces) to generate the pinion 46 and the hob 49 of the Gtype to generate the gear. It is desirable but not absolutely necessaryto make each hob of the same hand as the gear it is going to out.

A worm drive constructed on this principle is shown in Figures 12 and13. Now the worm 50 represents the rack P, Figure 5, and the worm wheel51 the pinion. The worm teeth 52 are of a convex outline conforming tothis theory. A concave contoured grinder 53 is used to grind the threadspreferably by the oscillating method as shown in my co-pendingapplication for patent, Serial No. 466,204, filed July '7, 1930. In apreferred construction I so select the radius of curvature b of the wormthread contour that the radii of curvature at the pitch point will beexactly the same in the worm and wheel. This is readily accomplished bysolving the Equation (6) for The said equation then becomes 2 1 Z a sin(21) b=2a sin Yo (22) that is, the radius of the wheel tooth flank isnow twice as great as in the common involute system.

Spiral bevel gears may also be improved upon this principle. In Figures14 and 15 the face mill cutter 54 having a convex tooth flank 55 of aradius 1) is used to generate the pinion, while the cutter 56 having aconcave tooth flank 57 of a radius 1) is used to generate the largernumber or the gear, of the drive.

What I claim as my invention is:

1. In a mating pinion and gear, a pair of conjugate tooth curves soformed that the tooth curve of the smaller member is the envelope of arack having convex tooth flanks when the pitch line of the said rackrolls over the pitch line of the said member and the tooth curve of thelarger member is the envelope of another rack having concave toothflanks, the second rack being an exact complement of the first rack andits curvature so determined that both generated cross contours remainconvex.

2. In a pinion a tooth curve obtained by scaling off a constant distancealong the normals of an extended involute.

3. In a gear a tooth curve obtained by scaling off a constant distancealong the normals of an abridged involute.

4. A pair of mating tooth curves in which the first curve is a parallelcurve to an extended involute generated from its pitch circle with apositive modification, and the second curve is a parallel curve to anabridged involute generated from the second pitch circle with a negativemodification and in which the modifications and the distances of thetooth curves from their respective base curves are respectively equal inabsolute value.

5. A cooperating rack and pinion in which the flanks of the rack teethare convex at all points thereof and the pinion flanks possess increasedradii of curvature as compared with the corresponding common involuteradii at all points thereof.

6. A pinion having an increased duration of contact permitting theemployment of fewer teeth than in standard construction in which thesaid increase is obtained by generating the teeth by means of a racktooth in which the pressure angle ever increases from the roots towardthe tips thereof in combination with a mating wheel in which thepressure angle of the generating rack tooth ever decreases from theroots to tips thereof. 1.

7. In a worm and wheel drive, a worm having a helical thread and convexthread contour as measured in its axial plane in which the radius ofcurvature of the said. contours is substantially twice as long as thecorresponding involute radius in the wheel tooth.

8. A spiral bevel pinion and gear in which the pinion teeth are such asmight be generated from a spiral crown wheel having convex tooth crosscontours and the teeth of the larger member from 1.

a similar crown wheel having concave tooth cross contours, thecorrection being such that both generated cross contours remain convex.

NIKOLA TRBOJEVICH.

